xf.kr 119
jpuk dh fofèk1. lqfoèkktud eki osQ ,d dkMZ cksMZ ij ,d pkVZ isij fpidkb,A
2. bl pkVZ isij ij ,d vkys[k dkX+k”k fpidkb,A
3. vkys[k dkX+k”k ij v{k X′OX vkSj YOY′ [khafp, (nsf[k, vko`Qfr 1)A
mís'; vko';d lkexzh
vkys[kh; fofèk ls foHkktu lw=k dklR;kiu djukA
dkMZ cksMZ] pkVZ isij] vkys[k dkX+k”k]xksan] T;kfefr ckWDl] isu @ isaflyA
fozQ;kdyki 11
vko`Qfr 1
4. bl vkys[k dkX+k”k ij nks fcanq A (x1, y
1) vkSj B (x
2, y
2) yhft, (nsf[k, vko`Qfr 2)A
5. fcanqvksa A vkSj B dks feykdj js[kk[kaM AB izkIr dhft,A
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120 iz;ksx'kkyk iqfLrdk
izn'kZu1. js[kk[kaM AB dks vkarfjd :i ls m : n osQ vuqikr esa fcanq C ij foHkkftr dhft,
(nsf[k, vko`Qfr 2)A2. vkys[k dkX+k”k ls fcanq C osQ funsZ'kkad if<+,A
3. foHkktu lw=k 2 1 2 1,
mx nx my nyx y
m n m n
+ += =
+ +dk iz;ksx djosQ] fcanq C osQ funsZ'kkad Kkr dhft,A
4. pj.k 2 vkSj pj.k 3 esa izkIr fd, x, C osQ funsZ'kkad ,d gh gSaAizs{k.k1. A osQ funsZ'kkad ___________ gSaA
B osQ funsZ'kkad ___________ gSaA2. fcanq C js[kk[kaM AB dks ___________ vuqikr esa foHkkftr djrk gSA3. vkys[k ls] C osQ funsZ'kkad ___________ gSaA4. foHkktu lw=k osQ iz;ksx ls C osQ funsZ'kkad ___________ gSaA5. vkys[k ls rFkk foHkktu lw=k ls izkIr C osQ funsZ'kkad ___________ gSaAvuqiz;ksxbl lw=k dk iz;ksx T;kfefr] lfn'k chtxf.kr rFkk f=kfoeh; T;kfefr esa fdlh f=kHkqt dk osaQnzdKkr djus esa fd;k tkrk gSA
vko`Qfr 2
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xf.kr 121
jpuk dh fofèk
1. lqfoèkktud eki dk ,d dkMZ cksMZ yhft, vkSj bl ij ,d pkVZ isij fpidkb,A
2. bl pkVZ isij ij ,d vkys[k dkX+k”k fpidkb,A
3. vkys[k dkX+k”k ij v{k X′OX vkSj YOY′ [khafp, (nsf[k, vko`Qfr 1)A
mís'; vko';d lkexzh
vkys[kh; fofèk ls f=kHkqt osQ {ks=kiQy osQlw=k dk lR;kiu djukA
dkMZ cksMZ] pkVZ isij] vkys[k dkX+k”k]xksan] isu @ isafly vkSj iVjhA
fozQ;kdyki 12
vko`Qfr 1
4. bl vkys[k dkX+k”k ij rhu fcanq A(x1,y
1)] B(x
2,y
2) vkSj C(x
3,y
3) yhft,A
5. bu fcanqvksa dks feykdj ,d f=kHkqt ABC izkIr dhft, (nsf[k, vko`Qfr 2)A
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122 iz;ksx'kkyk iqfLrdk
izn'kZu
1. lw=k] {ks=kiQy = ( ) ( ) ( )1 2 3 2 3 1 3 1 2
1– – –
2x y y x y y x y y + + dk iz;ksx djrs gq,]
∆ ABC dk {ks=kiQy ifjdfyr dhft,A
2. f=kHkqt ABC osQ vanj f?kjs gq, oxks± dh la[;k fuEufyf[kr izdkj ls fxudj f=kHkqt dk{ks=kiQy Kkr dhft,&
(i) ,d iw.kZ oxZ dks 1 oxZ yhft,A
(ii) vkèks ls vfèkd oxZ dks 1 oxZ yhft,A
(iii) vkèks oxZ dks 1
2 oxZ yhft,A
(iv) mu oxks± dks NksM+ nhft, tks vkèks ls de gSaA
3. lw=k }kjk ifjdfyr {ks=kiQy vkSj oxks± dh la[;k fxudj izkIr {ks=kiQy yxHkx cjkcjvFkkZr~ ,d gh gSa (nsf[k, pj.k 1 vkSj 2)A
vko`Qfr 2
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xf.kr 123
izs{k.k
1. A osQ funsZ'kkad ___________ gSaA
B osQ funsZ'kkad ___________ gSaA
C osQ funsZ'kkad ___________ gSaA
2. lw=k osQ iz;ksx ls] ∆ABC dk {ks=kiQy ___________ gSA
3. (i) iw.kZ oxks± dh la[;k = ___________ gSA
(ii) vkèks ls vfèkd oxks± dh la[;k = ___________ gSA
(iii) vkèks oxks± dh la[;k = ___________ gSA
(iv) oxks± dh la[;k fxudj izkIr oqQy {ks=kiQy = ___________ gSA
4. lw=k ls ifjdfyr {ks=kiQy vkSj oxks± dh la[;k fxudj izkIr {ks=kiQy ___________ gSaA
vuqiz;ksx
f=kHkqt osQ {ks=kiQy dk lw=k T;kfefr osQ vusd ifj.kkeksa dks Kkr djus esa mi;ksxh jgrk gS] tSlsfd rhu fcanqvksa dh lajs[krk dh tk¡p] f=kHkqt @ prqHkqZt @ cgqHkqt osQ {ks=kiQy ifjdfyr djukA
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124 iz;ksx'kkyk iqfLrdk
jpuk dh fofèk
I:
1. ,d jaxhu dkX+k”k @ pkVZ isij yhft,A blesa ls nks f=kHkqt ABC vkSj PQR ,sls dkVyhft, ftuesa laxr dks.k cjkcj gksa vFkkZr~ f=kHkqt ABC vkSj PQR esa] ∠A = ∠P;
∠B = ∠Q vkSj ∠C = ∠R gSA
mís'; vko';d lkexzh
nks f=kHkqtksa dh le:irk osQ fy, dlkSfV;k¡LFkkfir djukA
jaxhu dkX+k”k] xksan] LoSQp isu] dVj]T;kfefr ckWDlA
fozQ;kdyki 13
vko`Qfr 1 vko`Qfr 2
2. ∆ABC dks ∆PQR ij bl izdkj jf[k, fd 'kh"kZ A 'kh"kZ P ij fxjs rFkk Hkqtk AB HkqtkPQ osQ vuqfn'k jgs (Hkqtk AC Hkqtk PR osQ vuqfn'k jgs)] tSlk fd vko`Qfr 2 esa n'kkZ;kx;k gSA
izn'kZu I
1. vko`Qfr 2 esa] ∠B = ∠Q gSA D;ksafd laxr dks.k cjkcj gSa] blfy, BC||QR gSA
2. FksYl izes; (BPT) }kjk, PB PC AB AC
BQ CR BQ CR ;k
;kBQ CR
AB AC=
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xf.kr 125
;kBQ +AB CR +AC
AB AC= [nksuksa i{kksa esa 1 tksM+us ij]
;kAQ AR PQ PR
= =AB AC AB AC
;k ;k AB AC
=PQ PR
(1)
II:
1. ∆ABC dks ∆PQR ij bl izdkj jf[k, fd 'kh"kZ B 'kh"kZ Q ij fxjs rFkk Hkqtk BA HkqtkQP osQ vuqfn'k jgs (Hkqtk BC Hkqtk QR osQ vuqfn'k jgs)] tSlk fd vko`Qfr 3 esa n'kkZ;kx;k gSA
vko`Qfr 3
izn'kZu II
1. vko`Qfr 3 esa] ∠C = ∠R gSA D;ksafd laxr dks.k cjkcj gSa] vr% AC||PR gSA
2. FksYl izes; (BPT) }kjk, AP CR
AB BC= ; ;k
BP BR
AB BC= [nksuksa i{kksa esa 1 tksM+us ij]
;kPQ QR AB BC
AB BC PQ QR ;k (2)
(1) vkSj (2) ls, AB AC BC
PQ PR QR= =
bl izdkj] izn'kZuksa I vkSj II ls ge ikrs gSa fd tc nks f=kHkqtksa esa laxr dks.k cjkcj gksrsgSa] rks mudh laxr Hkqtk,¡ ,d gh vuqikr esa (vFkkZr~ lekuqikrh) gksrh gSaA vr%] nksuksaf=kHkqt le:i gksrs gSaA ;g f=kHkqtksa dh le:irk dh AAA dlkSVh gSA
14/04/18
126 iz;ksx'kkyk iqfLrdk
oSdfYid :i ls] vki f=kHkqt ABC vkSj PQR dh Hkqtk,¡ eki ldrs Fks rFkkfuEufyf[kr izkIr dj ldrs Fks&
AB AC BC
PQ PR QR= =
bl ifj.kke ls] ∆ABC vkSj ∆PQR le:i gSa] vFkkZr~ ;fn nks f=kHkqtksa esa rhuksa laxr dks.kcjkcj gksa] rks laxr Hkqtk,¡ lekuqikrh gksrh gSa vkSj blhfy, nksuksa f=kHkqt le:i gksrs gSaAblls f=kHkqtksa dh le:irk dh AAA dlkSVh izkIr gksrh gSA
III:
1. ,d jaxhu dkX+k”k @ pkVZ isij yhft, rFkk blesa ls nks f=kHkqt ABC vkSj PQR ,sls dkVyhft, fd budh laxr Hkqtk,¡ lekuqikrh gksa] vFkkZr~
vko`Qfr 4
AB BC AC
PQ QR PR= = gksaA
2. ∆ABC dks ∆PQR ij bl izdkj jf[k, fd 'kh"kZ A 'kh"kZ P ij fxjs rFkk Hkqtk AB HkqtkPQ osQ vuqfn'k jgsA è;ku nhft, fd Hkqtk AC Hkqtk PR osQ vuqfn'k jgrh gS(nsf[k, vko`Qfr 4)A
izn'kZu III
1. vko`Qfr 4 esa] AB AC
PQ PR= gSA blls
AB AC
BQ CR= izkIr gksrk gSA vr%] BC||QR gS
14/04/18
xf.kr 127
(FksYl izes; osQ foykse }kjk)] vFkkZr~] ∠B = ∠Q vkSj ∠C = ∠R gSA lkFk gh]∠A = ∠P gSA vFkkZr~ nksuksa f=kHkqtksa osQ laxr dks.k cjkcj gSaAbl izdkj] tc nks f=kHkqtksa dh laxr Hkqtk,¡ lekuqikrh gSa] rks muosQ laxr dks.k cjkcj gSaAblhfy,] nksuksa f=kHkqt le:i gSaA ;g nks f=kHkqtksa dh le:irk dh SSS dlkSVh gSA
oSdfYid :i ls] vki ∆ABC vkSj ∆PQR osQ dks.kksa dks eki dj ∠A = ∠P,
∠B = ∠Q vkSj ∠C = ∠R izkIr dj ldrs FksA bl ifj.kke ls] ∆ABC vkSj ∆PQR
le:i gSa] vFkkZr~ nks f=kHkqtksa esa rhuksa laxr Hkqtk,¡ lekuqikrh gksa rks laxr dks.k cjkcj gksrs gSa rFkkblhfy, f=kHkqt le:i gksrs gSaA blls nks f=kHkqtksa dh le:irk dh SSS dlkSVh izkIr gksrh gSA
IV:
1. ,d jaxhu dkX+k”k @pkVZ isij yhft, rFkk blesa ls nks f=kHkqt ABC vkSj PQR bl izdkj
vko`Qfr 5
dkV yhft, fd budh Hkqtkvksa dk ,d ;qXe lekuqikrh gks rFkk bu Hkqtkvksa osQ ;qXeksa osQvarxZr cus dks.k cjkcj gksaA
vFkkZr~] f=kHkqt ∆ABC vkSj ∆PQR esa] AB AC
PQ PR= rFkk ∠A = ∠P gksaA
2. ∆ABC dks ∆PQR ij bl izdkj jf[k, fd 'kh"kZ A 'kh"kZ P ij fxjs rFkk Hkqtk AB HkqtkPQ osQ vuqfn'k jgs] tSlk fd vko`Qfr 5 esa n'kkZ;k x;k gSA
izn'kZu IV:
1. vko`Qfr 5 esa] AB AC
PQ PR= gS] ftlls
AB AC
BQ CR= izkIr gksrk gSA
vr%] BC||QR (FksYl izes; osQ foykse ls) gS
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128 iz;ksx'kkyk iqfLrdk
vr%] ∠B = ∠Q vkSj ∠C = ∠R gSAbl izn'kZu ls] ge izkIr djrs gSa fd ,d f=kHkqt dh nks Hkqtk,¡ nwljs f=kHkqt dh nks HkqtkvksaosQ lekuqikrh gksa rFkk bu Hkqtkvksa osQ ;qXeksa osQ varxZr cus dks.k cjkcj gksa] rks nksuksa f=kHkqtksaosQ laxr dks.k cjkcj gSaA vr%] nksuksa f=kHkqt le:i gSaA ;g nks f=kHkqtksa dh le:irk dhSAS dlkSVh gSA
oSdfYid :i ls] vki ∆ABC vkSj ∆PQR dh 'ks"k Hkqtkvksa vkSj dks.kksa dks ekidj
∠B = ∠Q, ∠C = ∠R vkSj AB AC BC
PQ PR QR= = izkIr dj ldrs FksA
blls] ∆ABC vksj ∆PQR le:i gSa rFkk ge nks f=kHkqtksa dh le:irk dh SAS dlkSVhizkIr djrs gSaA
izs{k.k
okLrfod ekiu }kjk&
I. ∆ABC vkSj ∆PQR esa]
∠A = ______, ∠P = ______, ∠B = ______, ∠Q = ______, ∠C = ______,
∠R = ______,
AB
PQ = _______;
BC
QR = _________;
AC
PR = _________gSA
;fn nks f=kHkqtksa osQ laxr dks.k ___________ gSa] rks laxr Hkqtk,¡ ___________ gSaAvr%] f=kHkqt ___________ gSaA
II. ∆ABC vkSj ∆PQR esa]
AB
PQ = _______;
BC
QR = _________;
AC
PR = _________ gSA
∠A = _______, ∠B = _______, ∠C = _______, ∠P = _______,
∠Q = _______, ∠R = _______gSA
14/04/18
xf.kr 129
;fn nks f=kHkqtksa dh laxr Hkqtk,¡ ___________ gSa] rks muosQ laxr dks.k ___________
gksrs gSaA blhfy,] f=kHkqt ___________ gSaA
III. ∆ABC vkSj ∆PQR esa]
AB
PQ = _______;
AC
PR = _________
∠A = _______, ∠P = _______, ∠B = _______, ∠Q = _______,
∠C = _______, ∠R = _______gSA
tc ,d f=kHkqt dh nks Hkqtk,¡ nwljs f=kHkqt dh nks Hkqtkvksa osQ ______ gksa rFkk buosQvarxZr dks.k ______ gksa] rks f=kHkqt ______ gksrs gSaA
vuqiz;ksx
le:irk dh voèkkj.kk dk iz;ksx oLrqvksa osQ izfrfcacksa ;k fp=kksa dks NksVs lkb”k dk cukus ;kmudk vkoèkZu djus esa fd;k tkrk gSA
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130 iz;ksx'kkyk iqfLrdk
jpuk dh fofèk
1. ydM+h dh nks ifêð;k¡] eku yhft,] AB vkSj CD yhft,A
2. nksuksa ifêð;ksa dks fcanq O ij ijLij ledks.k ij izfrPNsn djrs gq, tksM+ nhft, (nsf[k,vko`Qfr 1)A
3. izR;sd iêðh ij cjkcj nwfj;ksa ij (O osQ nksuksa vksj) ik¡p dhysa yxkb, rFkk muosQ uke]eku yhft,] A
1, A
2, ......., A
5, B
1, B
2, ......., B
5, C
1, C
2, ......., C
5 vkSj D
1, D
2, .......,
D5 jf[k, (nsf[k, vko`Qfr 2)A
mís'; vko';d lkexzh
dhyksa rFkk nks izfrPNsnh ifêð;ksa dk iz;ksxdjrs gq,] le:i oxks± dk ,d fudk; [khapukA
nks ydM+h dh ifêð;k¡ (izR;sd dh pkSM+kbZ1cm vkSj yackbZ 30cm)] xksan] gFkkSM+k]dhysaA
fozQ;kdyki 14
vko`Qfr 1 vko`Qfr 2
4. nksuksa ifêð;ksa osQ pkj fljksa ij uhps fy[kh la[;k 1 okyh dhyksa (A1C
1B
1D
1) ij ,d èkkxk
yisfV, ftlls ,d oxZ izkIr gks tk, (nsf[k, vko`Qfr 2)A
5. blh izdkj] ifêð;ksa ij uhps fy[kh vU; leku la[;kvksa okyh dhyksa ij èkkxs yisfV,(nsf[k, vko`Qfr 3)A gesa oxZ A
1C
1B
1D
1 , A
2C
2B
2D
2, A
3C
3B
3D
3, A
4C
4 B
4 D
4 vkSj
A5C
5B
5D
5 izkIr gksrs gSaA
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xf.kr 131
izn'kZu
1. ifêð;ksa AB vkSj CD esa ls izR;sd ij dhysa lenwjLFk bl izdkj yxh gqbZ gSa fd
A1A
2 = A
2A
3 = A
3A
4 = A
4A
5, B
1B
2 = B
2B
3 = B
3B
4 = B
4B
5,
C1C
2 = C
2C
3 = C
3C
4 = C
4C
5, D
1D
2 = D
2D
3 = D
3D
4 = D
4D
5
2. vc fdlh ,d prqHkqZt eku yhft, A4C
4B
4D
4 esa (nsf[k, vko`Qfr 3)]
A4O = OB
4 = 4 bdkbZ
lkFk gh, D4O = OC
4 = 4 bdkbZ,
tgk¡ 1 bdkbZ nks ozQekxr dhyksa osQ chp dh nwjh gSA
vr%] fod.kZ ijLij lef}Hkkftr dj jgs gSaA
blfy,] A4C
4 B
4D
4 ,d lekarj prqHkZqt gSA
lkFk gh] A4B
4 = C
4D
4 = 4 × 2 = 8 bdkbZ gSA vFkkZr~ fod.kZ ijLij cjkcj gSaA
blosQ vfrfjDr] A4B
4 ⊥ C
4D
4 gS (D;ksafd ifêð;k¡ ledks.k ij izfrPNsn dj jgh gSa)
vr%] A4C
4B
4D
4 ,d oxZ gSA
blh izdkj] ge dg ldrs gSa fd A1C
1B
1D
1, A
2C
2B
2D
2, A
3C
3B
3D
3 vkSj A
5C
5B
5D
5
Hkh oxZ gSaA
vko`Qfr 3
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132 iz;ksx'kkyk iqfLrdk
3. vc oxks± dh le:irk dks n'kkZus osQ fy, (nsf[k, vko`Qfr 3)] A1C
1, A
2C
2, A
3C
3,
A4C
4, A
5C
5, C
1B
1, C
2B
2, C
3B
3, C
4B
4, C
5B
5 bR;kfn dks ekfi,A
lkFk gh] bu oxks± dh laxr Hkqtkvksa osQ vuqikr] tSls 2 2 2 2
3 3 3 3
A C C B,
A C C B, ... Hkh Kkr dhft,A
izs{k.k
okLrfod ekiu }kjk&A
2C
2 = ______, A
4C
4 = ______
C2B
2 = ______, C
4B
4 = ______
B2D
2 = ______, B
4D
4 = ______
D2A
2 = ______, D
4A
4 = ______.
2 2
4 4
A C
A C = _____,
2 2
4 4
C B
C B = _____,
2 2
4 4
B D
B D = _____,
2 2
4 4
D A
D A = _______gSA
lkFk gh] ∠A2 = _______, ∠C
2 = _______, ∠B
2 = _______,
∠D2 = _______, ∠A
4 = _______, ∠C
4 = _______,
∠B4 = _______, ∠D
4 = _______gSA
vr%] oxZ A2C
2B
2D
2 vkSj oxZ A
4C
4B
4D
4 ______gSaA
blh izdkj] izR;sd oxZ vU; oxZ osQ ______ gSA
vuqiz;ksx
le:irk dk mi;ksx oLrqvksa osQ izfrfcacksa dkvkoèkZu ;k mudk lkb”k NksVk djus esa (tSls,Vyl osQ ekufp=kksa esa) fd;k tkrk gS rFkk lkFkgh ,d gh usxsfVo ls fofHkUu lkb”kksa osQ I+kQksVkscukus esa Hkh bldk iz;ksx fd;k tkrk gSA
fVIi.kh
nksuksa fod.kks± dh yackb;k¡ vleku ysdjrFkk nksuksa ifêð;ksa osQ chp dk dks.k ledks.kls fHkUu ysdj] ge ;gh izfozQ;k viukrsgq,] le:i lekarj prqHkqZt@vk;r izkIrdj ldrs gSaA
lkoèkkfu;k¡
1- dhyksa vkSj gFkkSM+s dk iz;ksx djrsle; lkoèkkuh j[kuh pkfg,A
2- dhyksa dks cjkcj nwfj;ksa ij gh yxkukpkfg,A
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xf.kr 133
jpuk dh fofèk
1. ydM+h dh rhu ifêð;k¡ P, Q, vkSj R yhft, rFkk izR;sd iêðh dk ,d fljk vkoQfr 1 esan'kkZ, vuqlkj dkV yhft,A xksan@lsyksVsi dk iz;ksx djrs gq,] izR;sd iêðh osQ bu rhuksa fljksadks bl izdkj tksfM+, fd ;s lHkh fofHkUu fn'kkvksa esa jgsa (nsf[k, vkoQfr 1)A
2. izR;sd iêðh ij ik¡p dhysa yxkb, rFkk ifêð;ksa P, Q vkSj R ij yxh dhyksa osQ ozQe'k%uke P
1, P
2, ..., P
5, Q
1, Q
2, ..., Q
5 vkSj R
1, R
2, ..., R
5 nhft, (nsf[k, vko`Qfr 2)A
mís'; vko';d lkexzh
dhyksa lfgr Y osQ vkdkj dh ifêð;ksa dkiz;ksx djrs gq,] le:i f=kHkqtksa dk ,dfudk; [khapukA
cjkcj yackb;ksa (yxHkx 10cm yachvkSj 1cm pkSM+h) dh ydM+h dh rhuifêð;k¡] xksan] dhysa] lsyksVsi] gFkkSM+kA
fozQ;kdyki 15
vko`Qfr 1 vko`Qfr 2
3. rhuksa ifêð;ksa ij uhps fy[kh la[;k 1 okyh dhyksa ozQe'k% (P1, Q
1, R
1) ij èkkxk yisfV,
(nsf[k, vko`Qfr 3)A
4. vkSj vfèkd f=kHkqt izkIr djus osQ fy,] ozQe'k% ifêð;ksa ij uhps leku la[;k fy[kh dhyksaij èkkxs yisfV,A gesa f=kHkqt P
1Q
1R
1, P
2Q
2R
2, P
3Q
3R
3, P
4Q
4R
4 vkSj P
5Q
5R
5 izkIr
gksrs gSa (nsf[k, vko`Qfr 4)A
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134 iz;ksx'kkyk iqfLrdk
izn'kZu
1. rhuksa ydM+h dh ifêð;k¡ ,d fo'ks"k dks.k ij yxkbZ xbZ gSaA
vko`Qfr 3
vko`Qfr 4
2. ifêð;ksa P, Q, vkSj R esa ls izR;sd ij dhyssa cjkcj nwfj;ksa ij yxkbZ xbZ gSa] rkfd iêðhP ij P
1P
2=P
2P
3=P
3P
4=P
4P
5 gS rFkk blh izdkj ozQe'k% Q vkSj R ifêð;ksa ij
Q1Q
2=Q
2Q
3=Q
3Q
4=Q
4Q
5 vkSj R
1R
2=R
2R
3=R
3R
4=R
4R
5 gSA
3. vc dksbZ Hkh nks f=kHkqt] eku yhft,] P1Q
1R
1 vkSj P
5Q
5R
5 yhft,A Hkqtkvksa
P1Q
1, P
5Q
5, P
1R
1, P
5R
5, R
1Q
1 vkSj R
5Q
5 dks ekfi,A
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xf.kr 135
4. vuqikr 1 1
5 5
P Q
P Q,
1 1
5 5
P R
P R vkSj
1 1
5 5
R Q
R Q Kkr dhft,A
5. è;ku nhft, fd 1 1 1 1 1 1
5 5 5 5 5 5
P Q P R R Q
P Q P R R Q= = gSA
bl izdkj] ∆P1Q
1R
1 ~ P
5Q
5R
5 (SSS le:irk dlkSVh)
6. ;g ljyrk ls n'kkZ;k tk ldrk gS fd Y osQ vkdkj dh bu ifêð;ksa ij dksbZ Hkh nks f=kHkqtle:i gSaA
izs{k.k
okLrfod ekiu }kjk&P
1Q
1 = ______, Q
1R
1 = ______, R
1P
1 = ______,
P5Q
5 = ______, Q
5R
5 = ______, R
5P
5 = ______,
1 1
5 5
P Q
P Q = ______,
1 1
5 5
Q R
Q R = ______,
1 1
5 5
R P
R P = ______
vr%, ∆P1Q
1R
1 vkSj ∆P
5Q
5R
5 _______ gSaA
vuqiz;ksx
1. le:irk dh voèkkj.kk oLrqvksa osQ izfrfcacksa dk vkoèkZu djus ;k mudk lkb”k NksVk djus(tSls ,Vyl esa ekufp=k) esa iz;ksx dh tkrh gS rFkk lkFk gh ,d gh usxsfVo ls fofHkUulkb”kksa osQ I+kQksVks cukus esa Hkh iz;ksx dh tkrh gSA
2. le:irk dh voèkkj.kk dk iz;ksx djrs gq,] fdlh oLrq osQ vKkr ekiu mlh oLrq osQle:i oLrq osQ Kkr ekiuksa dh lgk;rk ls fuèkkZfjrfd, tk ldrs gSaA
3. le:irk dh voèkkj.kk dk iz;ksx djrs gq,] fdlhLraHk dh mQ¡pkbZ Kkr dh tk ldrh gS] ;fn mlLraHk dh lw;Z osQ izdk'k esa Nk;k dh yackbZ KkrgksA
fVIi.kh
mi;qDr dhyksa ij èkkxs yisVdj] ge ,slsle:i f=kHkqt Hkh izkIr dj ldrs gSa] tksleckgq u gksaA
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136 iz;ksx'kkyk iqfLrdk
mís'; vko';d lkexzh
vkèkkjHkwr lekuqikfrdrk izes; (FksYlizes;) dk lR;kiu djukA
gkMZ cksMZ ydM+h dh nks ifêð;k¡ (izR;sddh pkSM+kbZ 1cm vkSj yackbZ 10cm)]xksan] dVj] gFkkSM+k] dhysa] lI+ksQn dkX+k”k]f?kjfu;k¡] isap] LosQy] èkkxkA
fozQ;kdyki 16
vko`Qfr 1
jpuk dh fofèk1. lqfoèkktud eki dk gkMZ cksMZ dk ,d VqdM+k dkV yhft, vkSj ml ij ,d lI+ksQn dkX+k”k
fpidkb,Aèkkxk
isap osQ lkFk f?kjuhLosQy
isap osQ lkFk f?kjuh
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xf.kr 137
2. ydM+h dh nks iryh ifêð;k¡ yhft, ftu ij cjkcj nwfj;ksa ij 1, 2, 3, ... vafdr gks rFkkmUgsa ,d {kSfrt iêðh osQ nksuksa fljksa ij vko`Qfr 1 esa n'kkZ, vuqlkj mQèokZèkj :i ls yxknhft, rFkk bUgsa AC vkSj BD uke nhft,A
3. gkMZ cksMZ esa ls ,d f=kHkqtkdkj VqdM+k PQR dkV yhft, (eksVkbZ ux.; gksuh pkfg,) vkSjbl ij ,d jaxhu fpduk dkX+k”k fpidk yhft, rFkk bls lekarj ifêð;ksa AC vkSj BD osQchp esa bl izdkj jf[k, fd vkèkkj QR {kSfrt iêðh AB osQ lekarj jgs] tSls vkoQfr 1 esa[khapk x;k gSA
4. f=kHkqtkdkj VqdM+s dh vU; nks Hkqtkvksa ij vko`Qfr esa n'kkZ, vuqlkj 1] 2] 3--- la[;k,¡vafdr dhft,A
5. {kSfrt iêðh osQ vuqfn'k isap yxkb, rFkk cksMZ osQ mQijh Hkkx ij] fcanqvksa C vkSj D ijnks vkSj isap yxkb, rkfd A, B, D vkSj C ,d vk;r osQ 'kh"kZ cu tk,¡A
6. ,d :yj (LosQy) yhft, vkSj bl ij vko`Qfr esa n'kkZ, vuqlkj pkj Nsn dj yhft,rFkk bu Nsnksa ij isapksa dh lgk;rk ls pkj f?kjfu;k¡ yxk nhft,A
7. fcanq A, B, C vkSj D ij yxh dhyksa ls caèks èkkxs] tks f?kjfu;ksa ls mQij gksdj tk jgk gS]dk iz;ksx djrs gq,] cksMZ ij ,d LosQy yxk nhft, tSlk fd vko`Qfr 1 esa n'kkZ;k x;kgS] rkfd LosQy {kSfrt iêðh AB osQ lekarj f[kld losQ rFkk bls f=kHkqtkdkj VqdM+s ijLora=k :i ls mQij&uhps ljdk;k tk losQA
izn'kZu
1. LosQy dks mQèokZèkj ifêð;ksa ij] ∆ PQR osQ vkèkkj QR osQ lekarj j[krs gq,] eku yhft,fcanqvksa E vkSj F ij jf[k,A PE vkSj EQ rFkk lkFk gh PF vkSj FR nwfj;ksa dks ekfi,A
bldk ljyrk ls lR;kiu fd;k tk ldrk gS fd PE PF
EQ FR= gSA
blls vkèkkjHkwr lekuqikfrdrk izes; (FksYl izes;) dk lR;kiu gks tkrk gSA
2. LosQy dks ∆PQR osQ vkèkkj osQ lekarj mQij&uhps ljdkb, rFkk mijksDr fozQ;kdyki dksnksgjkb, vkSj LosQy dh fofHkUu fLFkfr;ksa osQ fy, FksYl izes; dk lR;kiu dhft,A
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138 iz;ksx'kkyk iqfLrdk
izs{k.k
okLrfod ekiu }kjk&PE = ___________, PF = ___________, EQ = ___________,
FR = ___________
PE
EQ = _______,
PF
FR= ___________ gSA
bl izdkj, PE PF
EQ FR= gSA blls izes; dk lR;kiu gks tkrk gSA
vuqiz;ksx
bl izes; dk mi;ksx f=kHkqtksa dh le:irk dh fofHkUu dlkSfV;ksa dks LFkkfir djus esa fd;k tkldrk gSA bldk mi;ksx ,d fn, gq, cgqHkqt osQ le:i] ,d fn, gq, LosQy xq.kd osQ lkFk],d vU; cgqHkqt dh jpuk djus esa Hkh fd;k tk ldrk gSA
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xf.kr 139
jpuk dh fofèk
1. ekiu 15 cm × 15 cm dk ,d jaxhu dkX+k”k yhft,A
2. ,d lI+ksQn dkX+k”k ij ,d f=kHkqt ABC [khafp,A
3. bl ∆ ABC dh ,d Hkqtk AB dks oqQN cjkcj Hkkxksa (eku yhft, 4 Hkkxksa) esa foHkkftrdhft,A
4. foHkktu osQ fcanqvksa ls gksdj] BC osQ lekarj js[kk[kaM [khafp, tks AC dks Øe'k% fcanqvksaE, H rFkk L ij izfrPNsn djrs gSa rFkk AC osQ foHkktu fcanqvksa ls gksdj] AB osQ lekarjjs[kk[kaM [khafp, (nsf[k, vko`Qfr 1)A
mís'; vko';d lkexzh
le:i f=kHkqtksa osQ {ks=kiQyksa vkSj Hkqtkvksa esalacaèk Kkr djukA
jaxhu dkX+k”k] xksan] T;kfefr ckWDl] oSaQph@]dVj] lI+ksQn dkX+k”kA
fozQ;kdyki 17
vko`Qfr 1
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140 iz;ksx'kkyk iqfLrdk
5. bl f=kHkqt dks jaxhu dkX+k”k ij fpidkb,A
6. ∆ ABC vc 16 lok±xle f=kHkqtksa esa foHkkftr gks x;k gS (nsf[k, vko`Qfr 1)A
izn'kZu
1. ∆AFH esa] 4 lok±xle f=kHkqt gSa rFkk blesa vkèkkj FH = 2DE gSA
2. ∆AIL esa] 9 lok±xle f=kHkqt gSa rFkk blesa vkèkkj IL = 3 DE = 3
2 FH gSA
3. ∆ABC esa] vkèkkj BC = 4DE = 2 FH = 4
3IL gSA
4. ∆ADE ~ ∆AFH ~ ∆AIL ~ ∆ABC
5.
2ar ( AFH) 4 FH
ar ( ADE) 1 DE
∆ = = ∆
2ar ( AIL) 9 IL
ar ( AFH) 4 FH
∆ = = ∆
2ar ( ABC) 16 BC
ar ( AFH) 4 FH
∆ = = ∆
vr%] le:i f=kHkqtksa osQ {ks=kiQyksa dk vuqikr mudh laxr Hkqtkvksa osQ oxks± osQ vuqikr osQcjkcj gksrk gSA
izs{k.k
okLrfod ekiu }kjk&BC = ______________, IL = ______________, FH = ______________,
DE = ______________.
eku yhft, fd ∆ADE dk {ks=kiQy 1 oxZ bdkbZ gSA rc]
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xf.kr 141
ar ( ADE)
ar ( AFH)
∆
∆ = __________,
ar ( ADE)
ar ( AIL)
∆
∆ = __________,
ar ( ADE)
ar ( ABC)
∆
∆ = __________,
2DE
FH
= __________,
2DE
IL
= __________,
2DE
BC
= __________
tks ;g n'kkZrs gSa fd le:i f=kHkqtksa osQ {ks=kiQyksa dk vuqikr mudh laxr Hkqtkvksa osQ oxks±osQ vuqikr osQ ______ gksrk gSA
vuqiz;ksx
;g ifj.kke nks le:i vko`Qfr;ksa osQ {ks=kiQyksa dh rqyuk djus esa mi;ksxh jgrk gSA
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142 iz;ksx'kkyk iqfLrdk
jpuk dh fofèk
1. lqfoèkktud eki dk ,d dkMZ cksMZ yhft, vkSj ml ij ,d lI+ksQn dkX+k”k fpidkb,A
2. jaxhu dkX+k”k ij x bdkbZ dh Hkqtk okyk ,d f=kHkqt (leckgq) cukb, vkSj bls dkV djfudky yhft, (nsf[k, vko`Qfr 1)A bls ,d bdkbZ f=kHkqt dfg,A
3. jaxhu dkX+k”kksa dk iz;ksx djrs gq,] mijksDr bdkbZ f=kHkqt osQ lok±xle i;kZIr la[;k esaf=kHkqt cukb,A
mís'; vko';d lkexzh
izk;ksfxd :i ls ;g lR;kfir djuk fd nksle:i f=kHkqtksa osQ {ks=kiQyksa dk vuqikr mudhlaxr Hkqtkvksa osQ oxks± osQ vuqikr osQ cjkcjgksrk gSA
jaxhu dkX+k”k] T;kfefr ckWDl] LoSQp isu]lI+ksQn dkX+k”k] dkMZ cksMZA
fozQ;kdyki 18
vko`Qfr 1 vko`Qfr 2
4. bu f=kHkqtksa dks dkMZ cksMZ ij vko`Qfr 2 vkSj vko`Qfr 3 esa n'kkZ, vuqlkj O;ofLFkr dhft,vkSj fpidkb,A
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xf.kr 143
izn'kZu
∆ABC vkSj ∆PQR le:i gSaA ∆ABC dh Hkqtk BC = (x + x + x + x) bdkbZ = 4x bdkbZ
∆PQR dh Hkqtk QR = 5x bdkbZ
∆ABC vkSj ∆PQR dh laxr Hkqtkvksa dk vuqikr
BC 4 4
QR 5 5
x
x= = gSA
∆ABC dk {ks=kiQy = 16 bdkbZ f=kHkqt
∆PQR dk {ks=kiQy = 25 bdkbZ f=kHkqt
∆ABC vkSj ∆PQR osQ {ks=kiQyksa dk vuqikr =
2
2
16 4
25 5= = ∆ABC vkSj ∆PQR dh laxr
Hkqtkvksa osQ oxks± dk vuqikr gSA
izs{k.k
okLrfod ekiu }kjk&x = ______, bdkbZ f=kHkqt (vko`Qfr 1 esa leckgq f=kHkqt) dk {ks=kiQy = _______
gSA
∆ABC dk {ks=kiQy = ______gS, ∆PQR dk {ks=kiQy = ______ gSA
vko`Qfr 3
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144 iz;ksx'kkyk iqfLrdk
∆ABC dh Hkqtk BC = ______gS, ∆PQR dh Hkqtk QR = ______ gSA
BC2 = ___________, QR2 = ___________,
AB = ___________, AC = ___________,
PQ = ___________, PR = ___________,
AB2 = ___________, AC2 = ___________,
PQ2 = ___________, PR2 = ___________,
2
2
BC
QR = ___________,
ABC
PQR
dk {k=s kiQydk {k=s kiQy
= ___________
2 22ABC BC AB –
PQR – – PR
dk {k=s kiQydk {k=s kiQy
vuqiz;ksx
;g ifj.kke f=kHkqtksa osQ vfrfjDr vU; le:ivko`Qfr;ksa osQ fy, Hkh iz;ksx fd;k tk ldrk gS]ftlls ckn esa Hkw[kaMksa] bR;kfn osQ ekufp=kksa dks rS;kjdjus esa lgk;rk feyrh gSA
fVIi.kh
;g fozQ;kdyki fdlh Hkh izdkj osQ f=kHkqt dksbdkbZ f=kHkqt ekudj fd;k tk ldrk gSA
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xf.kr 145
jpuk dh fofèk
1. ,d jaxhu pkVZ isij esa ls] fn, gq, prqHkqZt ABCD dk dVvkmV dkV yhft, rFkk blsvU; pkVZ isij ij fpidkb, (nsf[k, vko`Qfr 1)A
2. prqHkqZTk ABCD osQ vkèkkj (;gk¡ AB) dks vkarfjd :i ls] fn, gq, LosQy xq.kd ls izkIrvuqikr esa fcanq P ij foHkkftr dhft, (nsf[k, vko`Qfr 2)A
3. :yj (iVjh) dh lgk;rk ls prqHkqZt ABCD osQ fod.kZ AC dks feykb,A
4. P ls gksdj] ijdkj (lsV LDok;j ;k dkX+k”k eksM+us dh fozQ;k) dh lgk;rk ls js[kk[kaMPQ||BC [khafp,] tks AC ls R ij feys (nsf[k, vko`Qfr 3)A
mís'; vko';d lkexzh
,d fn, gq, LosQy xq.kd (1 ls de) osQvuqlkj] ,d fn, gq, prqHkqZt osQ le:iprqHkqZt dh jpuk djukA
pkVZ isij (jaxhu vkSj lI+ksQn)] T;kfefrckWDl] dVj] jcM+] MªkWbaxfiu] xksan] fiu]LoSQp isu] VsiA
fozQ;kdyki 19
vko`Qfr 1 vko`Qfr 2
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146 iz;ksx'kkyk iqfLrdk
5. R ls gksdj] ijdkj (lsV LDok;j ;k dkX+k”k eksM+us dh fozQ;k) dk iz;ksx djrs gq,] ,djs[kk[kaM RS||CD [khafp,] tks AD ls S ij feys (nsf[k, vko`Qfr 4)A
6. LoSQp isu dh lgk;rk ls prqHkqZt APRS esa jax Hkfj,A
APRS gh prqHkqZt ABCD osQ le:i fn, gq, LosQy xq.kd osQ vuqlkj ok¡fNr prqHkqZtgS (nsf[k, vko`Qfr 5)A
vko`Qfr 3 vko`Qfr 4
vko`Qfr 5
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xf.kr 147
izn'kZu
1. ∆ABC esa] PR||BC gSA vr%, ∆APR ~ ∆ABC gSA
2. ∆ACD esa] RS||CD gSA vr%, ∆ARS ~ ∆ACD gSA
3. pj.kksa 1 vkSj 2 ls] prqHkqZt APRS ~ prqHkqZt ABCD gSA
izs{k.k
okLrfod ekiu }kjk&AB = _____, AP = _____, BC = _____,
PR = _____, CD = _____, RS = _____,
AD = _____, AS = _____
∠A = _____, ∠B = _____, ∠C = _____,
∠D = _____, ∠P = _____, ∠R = _____,
∠S = _____ gSA
AP
AB= ______,
PR
BC= ______,
RS
CD= ______,
AS
AD= ______,
∠A = dks.k _____, ∠P = dks.k_____, ∠R = dks.k_____, ∠S = dks.k_____ gSAvr%] prqHkqZTk APRS vkSj ABCD _______ gSaA
vuqiz;ksx
bl fozQ;kdyki dk iz;ksx nSfud thou esa] ,d gh oLrq osQ fofHkUu lkbtksa esa fp=k (;k I+kQksVks)cukus esa fd;k tk ldrk gSA
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148 iz;ksx'kkyk iqfLrdk
jpuk dh fofèk
1. ,d fpduk dkX+k”k yhft, vkSj ml ij vkèkkj b bdkbZ vkSj yac a bdkbZ okyk ,dledks.k f=kHkqt [khafp,] tSlk vko`Qfr 1 esa n'kkZ;k x;k gSA
2. ,d vU; fpduk dkX+k”k yhft, vkSj ml ij vkèkkj a bdkbZ vkSj yac b bdkbZ okyk,d ledks.k f=kHkqt [khafp,] tSlk vko`Qfr 2 esa n'kkZ;k x;k gSA
mís'; vko';d lkexzh
ikbFkkxksjl izes; dk lR;kiu djukA pkVZ isij] fofHkUu jaxksa osQ fpdus (XysTM)dkX+k”k] T;kfefr ckWDl] oSaQph] xksanA
fozQ;kdyki 20
3. nksuksa f=kHkqtksa dks dkVdj fudky yhft, rFkk bUgsa ,d pkVZ isij ij bl izdkj fpidkb,fd nksuksa f=kHkqtksa osQ vkèkkj ,d gh ljy js[kk esa jgsa] tSlk vko`Qfr 3 esa n'kkZ;k x;k gSAbu f=kHkqtksa dks vko`Qfr esa n'kkZ, vuqlkj ukekafdr dhft,A
vko`Qfr 1 vko`Qfr 2
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xf.kr 149
4. CD dks feykb,A
5. ABCD ,d leyac gSA
6. ;g leyac rhu f=kHkqtksa APD, PBC vkSj PCD esa foHkkftr gks x;k gSA
izn'kZu
1. tk¡p dhft, fd ∆DPC dk ∠P ledks.k gSA
2. ∆APD dk {ks=kiQy 1
=2
ba oxZ bdkbZ
∆PBC dk {ks=kiQy 1
=2
ab oxZ bdkbZ
∆PCD dk {ks=kiQy 21
=2
c oxZ bdkbZ
3. leyac ABCD dk {ks=kiQy = ar(∆APD) + ar(∆PBC) + ar(∆PCD)
vr%, 21 1 1 1
2 2 2 2a b a b ab ab c
vko`Qfr 3
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150 iz;ksx'kkyk iqfLrdk
vFkkZr~, (a+b)2 = ab + ab + c2
vFkkZr~, a2 + b2 + 2ab = ab + ab + c2
vFkkZr~, a2 + b2 = c2
bl izdkj] ikbFkkxksjl izes; dk lR;kiu gks x;kA
izs{k.k
okLrfod ekiu }kjk& ∠ P = _______
AP = _______, AD = _______, DP = _______,
BP = _______, BC = _______, PC = _______gSA
AD2 + AP2 = _______, DP2 = _______,
BP2 + BC2 = _______, PC2 = _______gSA
bl izdkj, a2 + b2 = _______gSA
vuqiz;ksx
tc Hkh fdlh ledks.k f=kHkqt dh rhu Hkqtkvksa esa ls nks Hkqtk,¡ nh gksa] rks ikbFkkxksjl izes; }kjkrhljh Hkqtk Kkr dh tk ldrh gSA
14/04/18